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Ordinary Differential Equations

Tyn Myint-U

Chapter 2

First-Order Equations. Picard's Existence Theorem - all with Video Answers

Educators

Chapter Questions

01:26

Problem 1

Show that $\phi(x)=e^{x}$ is a solution of $y^{\prime \prime}=y^{\prime}$ on a certain interval $I$.

Alexandra Embry
Alexandra Embry
Numerade Educator
10:16

Problem 2

Verify that $\phi(x)=1 /(x+c)$ are solutions of $y^{\prime}=-y^{2}$ on certain intervals. Graph the solutions for $c=0, \pm 1, \pm 2$.

Pawan Yadav
Pawan Yadav
Numerade Educator
13:30

Problem 3

Draw the direction field for the equation $y^{\prime}=x^{2}+y^{2}$.

Chris Trentman
Chris Trentman
Numerade Educator
01:02

Problem 4

By the method of separation of variables solve $y^{\prime}=(\cos x)(y-2)$ satisfying the initial condition $y(0)=1$. Find the unique solution of the same equation satisfying $y(0)=2 .$ Specify the interval on which the solution is defined in each case.

Babita Kumari
Babita Kumari
Numerade Educator
07:10

Problem 5

Solve the equation
$$
y^{\prime}=1+y^{2}
$$
satisfying $y(0)=1 .$ Specify the interval on which the solution is defined.

Dwijendra Rao
Dwijendra Rao
Numerade Educator
02:52

Problem 6

Consider the initial-value problem
$$
\begin{array}{r}
y^{\prime}=3 y^{2 / 3}, \\
y\left(x_{0}\right)=y_{0}
\end{array}
$$
Discuss and sketch the solutions.

Taylor Shimono
Taylor Shimono
Numerade Educator
02:54

Problem 7

Solve the initial-value problem
$$
\begin{aligned}
y^{\prime} &=\alpha y-\beta y^{2}, \\
y(0) &=y_{0}
\end{aligned}
$$
where $\alpha$ and $\beta$ are small positive numbers. Show that
$$
\lim _{x \rightarrow \infty} \phi(x)=\left\{\begin{array}{ll}
\alpha / \beta & \text { for } y_{0}>0, \\
0 & \text { for } y_{0}=0 .
\end{array}\right.
$$
For $y_{0}<0, \phi$ is unbounded as $x$ approaches a certain value depending on $y_{0}$.

Minh Le
Minh Le
Numerade Educator
02:41

Problem 8

Solve the equation
$$
\left(x^{2}+y^{2}\right) y^{\prime}+x y=0
$$
by using the change of variable $v=y / x$ and $v=x / y$.

Brittany Knowlton
Brittany Knowlton
Numerade Educator
02:12

Problem 9

Show that the equation
$$
F(x, y)\left(x y^{\prime}-y\right)-f(x) y^{n}=0
$$
can be solved by the substitution $v=y / x$, where $F$ is homogeneous in $x$ and $y$.

Nick Johnson
Nick Johnson
Numerade Educator
02:29

Problem 10

Solve the linear fractional equation
$$
y^{\prime}=\frac{a x+b y}{c x+d y}, \quad a d \neq b c
$$
Also obtain solutions by using polar coordinates.

Carson Merrill
Carson Merrill
Numerade Educator
06:33

Problem 11

Test for exactness and solve the following equations.
(a)
$$
\left(\frac{1}{x}-\frac{1}{y}\right) d x+\frac{x}{y^{2}} d y=0
$$
(b) $(y \cos x-\sin y) d x+(\sin x-x \cos y) d y=0$
(c) $(1-x y)^{-2} d x+\left[y^{2}+x^{2}(1-x y)^{-2}\right] d y=0$

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
01:43

Problem 12

If $M=y f(x y)$ and $N=x g(x y)$, show that $1 /(M x-N y)$ is an integrating factor provided $M x-N y \neq 0 .$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
04:12

Problem 13

Show that $1 / x^{2}, 1 / y^{2}, 1 / x y, 1 /\left(x^{2} \pm y^{2}\right)$ are the integrating factors of the equation $y^{\prime}=y / x$, and obtain the solutions representing the same family of curves.

Diego Rojas
Diego Rojas
Numerade Educator
02:29

Problem 14

Show that the solution of the initial-value problem
$$
\begin{array}{r}
y^{\prime}-2 x y=1 \\
y(0)=1
\end{array}
$$
is
$$
\phi(x)=e^{x^{2}}\left(1+\frac{\sqrt{\pi}}{2} \operatorname{erf}(x)\right)
$$
where the emor function is defined by
$$
\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t
$$
and is tabulated.

Linda Hand
Linda Hand
Numerade Educator
05:49

Problem 15

Describe the behavior of the solution of
$$
y^{\prime}+\frac{1}{x} y=\frac{\cos x}{x}
$$
as $x \rightarrow 0$.

Willis James
Willis James
Numerade Educator
00:47

Problem 16

A function is said to be periodic with period $p$ if
$$
f(x+n p)=f(x)
$$
where $n$ is an integer. Suppose that $f(x)$ is continuous and periodic with period $p$ for all $x$. Show that if $\phi(x)$ is a solution of the homogeneous equation
$$
y^{\prime}+f(x) y=0
$$
then $\phi(x+p)$ is also a solution. Show that for some constant $c$,
$$
\phi(x+p)=c \phi(x)
$$
for all $x$.

Linh Vu
Linh Vu
Numerade Educator
08:30

Problem 17

Consider the initial-value problem
$$
\begin{aligned}
y^{\prime} &=x+y \\
y(0) &=0
\end{aligned}
$$
By the Picard method of successive approximations, show that the successive approximations $\phi_{0}, \phi_{1}, \ldots, \phi_{n}, \ldots$ exist for all $x .$ Compute $\phi_{0}, \phi_{1}, \phi_{2}, \phi_{3}, \phi_{4}$ and compare with the exact solution.

Brittany Knowlton
Brittany Knowlton
Numerade Educator
07:10

Problem 18

Show that a unique solution exists for each of the following initial-value problems.
(a)
$$
\begin{aligned}
y^{\prime} &=x^{2}+y^{2} \\
y(0) &=0
\end{aligned}
$$
on the interval $|x|<\frac{1}{\sqrt{2}}$.
(b)
$$
y^{\prime}=1+y^{2}
$$
$$
y(0)=0
$$
on the interval $|x|<\frac{1}{2}$.

Dwijendra Rao
Dwijendra Rao
Numerade Educator
01:07

Problem 19

(Peano's existence theorem). Suppose $f(x, y)$ is continuous on the rectangle $\left|x-x_{0}\right| \leqslant h,\left|y-y_{0}\right| \leqslant k$, where $|f(x, y)| \leqslant M$. Prove that there exists at least one solution $\phi$ such that $\phi^{\prime}=f(x, \phi)$ on the interval $\left|x-x_{0}\right|<\min (h, k / M)$, satisfying the initial condition $y\left(x_{0}\right)=y_{0}$.

Linh Vu
Linh Vu
Numerade Educator
01:03

Problem 20

An equation of the form
$$
y=x p+f(p)
$$
where $p=d y / d x$, is known as a Clairaut equation.
(a) Differentiate (A) with respect to $x$ and obtain
$$
\left[x+f^{\prime}(p)\right] \frac{d p}{d x}=0
$$
(b) If $\left[x+f^{\prime}(p)\right] \neq 0$, then $d p / d x=0$, which gives $p=c$, and as a result $y=c x+f(c)$
(c) If $x+f^{\prime}(p)=0$, then a singular solution is obtained by eliminating
$p$ between the equations $x+f^{\prime}(p)=0$ and $y=x p+f(p)$.
(d) Determine a one-parameter family of solutions of
$$
y=x p+p^{2} .
$$
Find the singular solution also.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
06:32

Problem 21

Any first-order equation
$$
y^{\prime}=a(x)+b(x) y+c(x) y^{2}
$$
is called a Riccati equation. Suppose that a particular solution $y_{1}(x)$ of this equation is known.
(a) By the substitution $y(x)=y_{1}(x)+1 / u(x)$, show that the Riccati equation is transformed into a linear first-order equation
$$
\frac{d u}{d x}=-\left(b+2 c y_{1}\right) u-c
$$
(b) If the substitution is $y(x)=y_{1}(x)+u(x)$, show that Riccati's equation is reduced to a Bernoulli equation.
(c) Solve $y^{\prime}=1+x^{2}-2 x y+y^{2}$ if $y_{1}(x)=x$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator